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    Simple Type Theory (STT) gives rise to Russell's Appendix... — Carmelics
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    Simple Type Theory (STT) gives rise to Russell's Appendix B paradox when supplemented with the principle that propositions differing by a constituent are distinct propositions and a correlation of propositions with classes they mention.

    Modality & PossibilityPhilosophy of Language
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    1 reason for
    2 reasons against

    Reasons For

    1 perspective
    Reason for
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    • 1.If two propositions differ by a constituent, then they are different propositions (the Structure principle).
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    • 2.If propositions are correlated with the classes they mention, then diagonalization is possible.
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    • 3.Let W be the class of propositions not in the classes with which they are correlated; the proposition 'Every proposition in W is true' is in W if and only if it is not in W.
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    Reasons Against

    2 perspectives
    Reason against 1 of 2
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    • 1.The 'Structure principle' that propositions differing by a constituent are distinct conflates syntactic individuation with intensional identity of structured propositions.
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    • 2.Church's own Logic of Sense and Denotation (Alternative (0)) individuates propositions by synonymous isomorphism, not mere constituent difference, blocking the diagonal construction.
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    • 3.If propositions are individuated by sense-equivalence rather than syntactic structure, the correlation W cannot be well-defined, since 'mentioning' a class is not a purely extensional relation.
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    Reason against 2 of 2
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    • 1.Quine argued in 'Set Theory and Its Logic' that Russell-style paradoxes generated by type theory reveal the instability of treating propositions as set-like objects subject to membership conditions.
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    • 2.If propositions are not genuine objects in the domain over which quantifiers range—as deflationary nominalists like Prior urged—then no well-formed correlation between propositions and classes exists to generate the diagonal.
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    • 3.Prior's substitutional account of propositional quantification in 'Objects of Thought' shows STT's paradox depends on a realist ontology of propositions that is independently contestable.
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    Topics

    Philosophy of LanguageModality & Possibility

    Key Terms

    Appendix B paradox(as a named logical problem in Russell's work)
    A specific logical contradiction that Russell discovered, arising from certain assumptions about how types and propositions work together.
    Classes(as mathematical/logical groupings)
    Groups or sets of objects that share something in common; a way of collecting similar things together.
    Russell
    # Russell Russell most commonly refers to **Bertrand Russell**, a highly influential British philosopher, logician, and social critic (1872-1970) who fundamentally changed how we think about logic, language, and knowledge. He's famous for showing that common-sense reasoning can contain hidden contradictions and for arguing that philosophy should use the precision of mathematics to solve problems. Russell also became a prominent public intellectual who wrote about everything from religion to nuclear weapons, making him one of the most important thinkers of the 20th century.
    Simple Type Theory (STT)(Contrasted with Ramified Type Theory as a framework that does not block the paradox)
    A type-theoretic logical system in which the Appendix B paradox can be derived when supplemented with the Structure principle and a predicate for identity of propositions.
    constituent(Russellian/direct reference theories of propositions)
    An entity that is literally part of a proposition, as the individual themselves is part of a singular proposition
    paradox(R. M. Sainsbury's definition, presented as a target of criticism)
    An apparently unacceptable conclusion derived by apparently acceptable reasoning from apparently acceptable premises
    proposition(Used in the context of a semantic theory sensitive to differences in subject matter.)
    The content expressed by a sentence, individuated at least in part by the subject matter of the sentence and the contents of its subsentential expressions.

    Related

    Church's own Logic of Sense and Denotation (Alternative (0)) individuates propos...If propositions are correlated with the classes they mention, then diagonalizati...If propositions are individuated by sense-equivalence rather than syntactic stru...If propositions are not genuine objects in the domain over which quantifiers ran...
    +5 moreShow less
    If two propositions differ by a constituent, then they are different proposition...Let W be the class of propositions not in the classes with which they are correl...

    Source

    AI-extracted1/3 agreementValid
    SEP: church
    View source passageHide passage
    In a later paper, Church turns his attention to a paradox that Russell formulates in Appendix B of Principles of Mathematics (1903). Once again, Church’s aim is first to show how this contradiction arises in STT, and then to appeal to RTT as a means of solving the problem. This time, however, STT is not supplemented directly by any overt semantic principles concerning denotation or predication, but only by a predicate for identity of propositions and axioms supporting Russell’s view as of 1903,
    Extraction notes

    Validity: Extracted via Max plan + API grounding/validity checks

    Details

    Prior's substitutional account of propositional quantification in 'Objects of Th...
    Quine argued in 'Set Theory and Its Logic' that Russell-style paradoxes generate...
    The 'Structure principle' that propositions differing by a constituent are disti...

    Similar

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